12,529 research outputs found
On the Argument from Physics and General Relativity
I argue that the best interpretation of the general theory of relativity has need of a causal entity, and causal structure that is not reducible to light cone structure. I suggest that this causal interpretation of GTR helps defeat a key premise in one of the most popular arguments for causal reductionism, viz., the argument from physics
Formalizing Mathematical Knowledge as a Biform Theory Graph: A Case Study
A biform theory is a combination of an axiomatic theory and an algorithmic
theory that supports the integration of reasoning and computation. These are
ideal for formalizing algorithms that manipulate mathematical expressions. A
theory graph is a network of theories connected by meaning-preserving theory
morphisms that map the formulas of one theory to the formulas of another
theory. Theory graphs are in turn well suited for formalizing mathematical
knowledge at the most convenient level of abstraction using the most convenient
vocabulary. We are interested in the problem of whether a body of mathematical
knowledge can be effectively formalized as a theory graph of biform theories.
As a test case, we look at the graph of theories encoding natural number
arithmetic. We used two different formalisms to do this, which we describe and
compare. The first is realized in , a version of Church's
type theory with quotation and evaluation, and the second is realized in Agda,
a dependently typed programming language.Comment: 43 pages; published without appendices in: H. Geuvers et al., eds,
Intelligent Computer Mathematics (CICM 2017), Lecture Notes in Computer
Science, Vol. 10383, pp. 9-24, Springer, 201
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
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